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Theorem ovresd 7574
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (𝜑𝐴𝑋)
ovresd.2 (𝜑𝐵𝑋)
Assertion
Ref Expression
ovresd (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (𝜑𝐴𝑋)
2 ovresd.2 . 2 (𝜑𝐵𝑋)
3 ovres 7573 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 585 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   × cxp 5675  cres 5679  (class class class)co 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  sscres  17770  fullsubc  17800  fullresc  17801  funcres2c  17852  psmetres2  23820  xmetres2  23867  prdsdsf  23873  xpsdsval  23887  xmssym  23971  xmstri2  23972  mstri2  23973  xmstri  23974  mstri  23975  xmstri3  23976  mstri3  23977  msrtri  23978  tmsxpsval  24047  ngptgp  24145  nlmvscn  24204  nrginvrcn  24209  nghmcn  24262  cnmpt1ds  24358  cnmpt2ds  24359  ipcn  24763  caussi  24814  causs  24815  minveclem2  24943  minveclem3b  24945  minveclem3  24946  minveclem4  24949  minveclem6  24951  ftc1lem6  25558  ulmdvlem1  25912  abelth  25953  cxpcn3  26256  rlimcnp  26470  hhssnv  30517  madjusmdetlem3  32809  qqhcn  32971  qqhucn  32972  ftc1cnnc  36560  ismtyres  36676  isdrngo2  36826  naddcnffo  42114  rngchom  46865  ringchom  46911  irinitoringc  46967  rhmsubclem4  46987
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